Towards Modeling Trust Based Decisions: A
Game Theoretic Approach
Vidyaraman Sankaranarayanan, Madhusudhanan Chandrasekaran ,
and Shambhu Upadhyaya
University at Buffalo, Buffalo NY 14260, USA
Phone.: 716-645-3180, Fax.: 716-645-3464
{vs28,mc79,shambhu}@cse.buffalo.edu
Abstract. Current trust models enable decision support at an implicit
level by means of thresholds or constraint satisfiability. Decision support
is mostly included only for a single binary action, and does not explicitly consider the purpose of a transaction. In this paper, we present a
game theoretic model that is specifically tuned for decision support on a
whole host of actions, based on specified thresholds of risk. As opposed
to traditional representations on the real number line between 0 and +1,
Trust in our model is represented as an index into a set of actions ordered according to the agent’s preference. A base scenario of zero trust
is defined by the equilibrium point of a game described in normal form
with a certain payoff structure. We then present the blind trust model,
where an entity attempts to initiate a trust relationship with another
entity for a one-time transaction, without any prior knowledge or recommendations. We extend this to the incentive trust model where entities
can offer incentives to be trusted in a multi-period transaction. For a
specified risk threshold, both models are analyzed by using the base scenario of zero trust as a reference. Lastly, we present some issues involved
in the translation of our models to practical scenarios, and suggest a rich
set of extensions of the generalized game theoretic approach to model
decision support for existing trust frameworks.
Keywords: Decision Support, Game Theory, Incentives, Risk, Trust.
1
Introduction
The three dominant characteristics of trust are vulnerability, risk and expectation
(or uncertainty). All trust models encompass these characteristics and present
definitions, representations, evaluations and operations on the notion of trust
(see [1,2,3,4,5] and the references therein). Decision support for trust models
and frameworks must involve an accurate estimation of the uncertainty of other
agent’s actions. The level to which an agent is willing to tolerate the loss due
to the uncertainty is the risk threshold. Most trust models/frameworks enable
decision support based on threshold values or constraint satisfiability (e.g., automated trust negotiations first initiated by Winsborough et al. [6] and later
extended in [7,8,9]) or some aggregation metric of recommendations (e.g., Fuzzy
J. Biskup and J. Lopez (Eds.): ESORICS 2007, LNCS 4734, pp. 485–500, 2007.
c Springer-Verlag Berlin Heidelberg 2007
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S. Vidyaraman, M. Chandrasekaran, and S. Upadhyaya
metrics [10]) based on past history, like recommender systems (RS) [11]. In most
of the previous works that present generic trust models, the decision making
criteria, i.e., the translation from trust to action, is left to the agent, and rightly
so, because such translations are usually context dependent. Decision support
is not explicitly embedded into the trust model; rather the agent is expected to
make decisions for a single action based on thresholds or constraints, depending
on the model. Such threshold or constraint specification is for a single binary
action and is not applicable when the agent has a multitude of action choices.
The traditional trust values of 0 to +1 are not particularly conducive towards
the direct translation of trust to a multitude of actions; an additional mapping
function is required for decision support.
In this paper, we present a model of trust based decisions using a game theoretic approach. In our model, agents have a multitude of action choices and
interact with other agents with some (possibly zero) trust. The trust of an agent
(on other agents) is represented as an index into the action set of the agent.
Thus, the very value of trust enables decision support, even at the level of the
abstract model. In other words, the trust value describes the action to be initiated in an interaction. We extend the work of other trust models by going
beyond defining trust notions; taking our intuition from automated trust negotiations [6], our model assumes that every trust interaction has a purpose, and
thus, both (and in general all ) interacting agents must have something to gain
at the end of an interaction. With this purpose in mind, we first present a base
scenario/game where interacting agents do not trust each other, and thus play
at their equilibrium point in game theoretic terms. We then present the basic
blind trust model, which is a one-time transaction between two agents. Here,
an agent is assumed to trust another agent for reasons outside the scope of the
model; no assumptions on the application domain are made, neither are reasons
for trusting an agent provided. At this point, we define the notion of a desired
action (in game theoretic terms) that formalizes the expectation of a trusting
agent. We then define two types of risk in the blind trust model and evaluate
the number of rounds of sequential game play a trusting agent (or truster) may
expect to play for a given risk tolerance. Next we consider the purpose of a trust
interaction and present the incentive trust model, where agents can provide an
incentive to other agents in order to adhere to a minimal trust level that is established or agreed upon in advance. The incentive trust model also provides
an advantage to a newly entering agent in the transaction who has no history;
instead of starting from a default low trust value, the agent may quickly build
up its reputation by offering incentives. In both these models, we use the base
scenario as a reference point for determining the loss of an agent when trust is
misplaced or violated. This loss is used to derive metrics for estimating the risk
faced by an agent. Finally, we present a rich set of models and frameworks to
which the general game theoretic approach may be extended. To the best of our
knowledge, such explicit decision support and analysis in trust models through
a game theoretic approach has not been done so far.
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In this paper, we do not explicitly define the notions of trust (and distrust),
(atomic) purpose of the trust relationship, etc. These notions have been well
defined in previous works [4,1,5] (also see [3,2] and the references therein for
a listing of previous works and models); indeed, the core notion of trust could
stem from any of the prior works that not only define fundamental concepts
well, but also provide means for evaluating and performing other operations (like
comparison) on trust through the history of past transactions. Unlike previous
approaches, the purpose of this work is to provide a game theoretic model of
trust based decisions. This work represents a natural progression of existing
trust models to provide explicit decision support for agents with a multitude of
action choices. By proper mechanism design [12,13], game theoretic models can
also subsume other models to provide a well analyzed decision support theory.
1.1
Summary of Contributions
The contributions of this paper are in the theoretical realm of trust and are
summarized as follows.
1. We present a game theoretic model for enabling trust based decision support
by defining trust as an index into the action set of a trusting agent.
2. A blind trust model is presented, where agents engage in repeated games;
two types of risk are defined and the number of rounds an agent may expect
to play is analyzed for a given risk threshold.
3. An incentive trust model, where all interacting agents stand to gain at the
end of the transaction is presented.
(a) Sufficient conditions are derived for both the agent offering the incentive and the trusting agent/truster in order for the interaction to be
successful.
(b) This model offers a mechanism for a new agent to start an interaction
with a high level of trust instead of the default low value by offering
incentives.
4. We present directions for the translation of the game theoretic models for
practical applications and suggest potentially rich areas of future works.
The rest of this paper is organized as follows. Section 2 describes the related
work; Section 3 presents a brief background and intuitive description of the game
theoretic approach of our model. Section 4 presents the base scenario of zero
trust, the blind trust and incentive trust models. Section 5 presents some of the
issues involved in applying the game theoretic models to practical applications.
Concluding remarks and directions for future works are presented in Section 6.
2
Related Work
The work in this paper primarily focuses on enabling decision support for agents
operating on the notion of trust. By its very definition [1,2,3,4,5], trust implies
a certain amount of risk due to uncertainty in the interacting agents decision
criteria. Trust and risk have both been used to decide or optimize the effective
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S. Vidyaraman, M. Chandrasekaran, and S. Upadhyaya
payoff [14] or lower the expected loss [15]. Their relationship towards decision
support has been investigated in [16]. An important factor of reciprocity in terms
of trust has been experimentally investigated in [17].
Game theoretic descriptions and analysis of trust have been investigated by
[18]; some games that cannot be represented in a normal form have been investigated experimentally [19]. Decision support based on trust has been investigated
for electronic transactions [20], trust negotiations [7,8,21,9,6], etc. In fact, almost
all trust models incorporate some decision criteria at least implicitly; however
most of them are binary or threshold based, in that an action may be initiated
if a certain constraint is satisfied or the trust value is above a certain threshold. In
this work, we explicitly consider the purpose of any trust transaction and assume
that all interacting entities gain at the end of the transaction. Our work is similar to game theoretic modeling of an auction marketplace, where agents choose
actions with optimal payoffs. The work by Lam et al. [14] discusses trade in open
marketplaces using trust and risk, and is the closest to the work in this paper.
Our work substantially builds on the work of previous trust models by providing
a model for trust based decision support, particularly in situations where an agent
has a multitude of actions to choose from; agents can not only decide which action to take based on their trust value, but also evaluate the number of rounds of
interaction (game) they can engage for a given threshold of risk.
3
Background and Overview
In order to make this paper self sufficient, we first give a brief descriptive background of a 2-player game and its corresponding Nash Equilibrium. We then
present an overview of our notion of trust in a 2-player game. A 2-player play
game is defined as a game between the players denoted as P1 and P2 . Each
player is required to operate simultaneously over an action space. On the completion of an action, both players receive a payoff or a reward depending on the
actions chosen by themselves and the other player. Games where the players
have opposing goals are called non-cooperative games. The goal of each player
is to maximize his or her own payoff. Towards this goal, each player develops
a strategy over his/her actions spaces, thereby ensuring the best payoff in the
game. Assuming that both players know each others action space and their corresponding payoffs, their strategy will be to choose the best response action for the
other player’s best strategy. Furthermore, assuming both players are perfectly
rational, i.e., they both can efficiently compute each others best strategy recursively, their final action will be one from which each can hope to gain nothing by
deviating unilaterally. Intuitively, such a ‘final’ action results in ‘equilibrium’ for
both players; in order to maximize their payoffs, each player only need to play
that particular action, regardless of how many times the game is played. Such
an action profile is called a Nash Equilibrium.
In the interest of keeping the overview concise and intuitive, a number of
details have been omitted; e.g., not all games have a single action strategy that
results in equilibrium; there is usually a probability distribution on the action
Towards Modeling Trust Based Decisions: A Game Theoretic Approach
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space that also results in equilibrium (such an action profile is called a mixed
strategy, while the previous single action equilibrium is called a pure strategy).
The reader is referred to [22,23] for a more detailed exposition on game theory.
We now describe the incorporation of trust into a standard 2-player non-zero
sum game. In any game in the equilibrium situation, players have no incentive to
deviate from their chosen actions. We formulate our model as follows. Consider
the players (P1 , P2 ) whose equilibrium actions are (a1 , b1 ) with a payoff of (50,
50). If P1 were to choose some other action a2 , then in a typical game, there exists
a best response action b2 for player P2 such that the effective payoff is (40,100), i.e.,
P2 ’s payoff would be larger than the equilibrium payoff and P1 ’s payoff would be
smaller (maybe negative too, but we assume positive payoffs). Now assume that
there exists an action b3 , called the desired response, for player P2 such that the
payoff is (60, 70), i.e., both players stand to gain from the equilibrium payoff, but
player P2 stands to gain lesser that the optimum best response for action a2 (which
happens to be action b2 ). Thus, player P1 is said to trust player P2 , if on playing
a2 , there is an expectation that P2 would respond with b3 instead of b2 , thereby
leading to a profit on both sides, but not necessarily the maximum allowable for
player P2 . Now imagine a continuum of such actions ak , ak+1 , etc., for P1 such that
P2 can respond with actions bk , bk+1 , etc., such that their payoffs are increasingly
better than the equilibrium payoff, but P2 ’s payoff is lesser that the best response
actions bbest−response to ak , ak+1 , etc. Then P1 ’s trust in P2 is the index k into his
action profile. If the action profiles are suitably ordered, an increasing index value
indicates an increasing level of trust.
The intuition behind our model is simple: an act of trust implies, amongst
other things, (a) a potential vulnerability on the part of the trusting agent, (b) a
threshold of risk the trusting agent is willing to tolerate and (c) an uncertainty
(or expectation) on the response of the other agent: i.e., the three dominant characteristics, vulnerability, risk and expectation/uncertainty have to be embedded
into the model. The vulnerability on the part of the trusting agent is expressed
by its deviation from the equilibrium play. The extent to which the trusting
agent is willing to expose itself to the vulnerability is the risk. The responding
agent may initiate the best response (and hence violate the trust placed in him)
or initiate the desired response; thus, there is an uncertainty on the response
type. The equilibrium point/action profile forms the Base Scenario, where the
players do not trust each other.
3.1
Blind Trust and Incentive Trust Models
The blind trust model presents the scenario where an agent trusts another agent
with no assurances or guarantees. In this model, the vulnerability faced by an
agent when trusting another agent is expressed in terms of the possible loss
of payoff for one round. Then, given the risk an agent is willing to take (the
maximum vulnerability), we analyze the number of rounds of the game the player
may expect to play before getting back to the equilibrium play. Thus far, the trust
of a single player is unconditional; we now introduce the incentive trust model
where users can trade a predefined amount of their payoffs before the initiation
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of single round of the game. We now consider the interactions between players
and their decision criteria when there is an expectation of a return at a later
point in time, i.e., there is a purpose to the entire transaction and all involved
entities expect to gain something at the end of the transaction. Many trust
relationships fall under this category. Consider a customer who interacts with a
service provider; the customer may pay a premium for a service that he expects at
a later time. Thus, the customer can be said to trust the service provider to keep
his end of the contract. Such a scenario is also applicable in the security field, in
Automated Trust Negotiation mechanisms and protocols. Trust Negotiation was
first introduced by Winsborough [6] as a means for agents to negotiate their trust
with other agents in a heterogeneous environment. A commonly quoted example
is the interaction between a potential customer Alice and a service provider Bob.
The simplified scenario is as follows: a customer Alice wishes to make a purchase
from Bob, a service provider, but is initially unwilling to provide any means of
authentication (Drivers License, Credit Card Number, etc.). Bob provides Alice
with a certificate from the Better Business Bureau (BBB) stating that he is
indeed a service provider with a certain standing. The BBB certificate is verified
by Alice and it ‘helps’ her to make a trust decision about Bob; she provides
her Resellers License/Credit Card number to Bob to make a purchase. Bob
verifies her license/CC number and completes the transaction. This example
is illustrative of the general trust transaction: (a) there is a purpose to the
transaction and (b) both (and in general all) agents stand to gain at the end of
the transaction. We analyze these scenarios and present decision making criteria
for specified risk thresholds.
3.2
A Note on Mechanism Design
Finally, we conclude with a note on the concept of mechanism design [12,13] in
game theory. Loosely speaking, mechanism design can be viewed as a technique
for designing a game so that rational and selfish agents do not have an incentive
to deviate from the desired behavior of the game designer. Proper mechanism
design maps the desired behavior of agents to the equilibrium play so that no
agent can gain by deviating from the equilibrium point. From a purely game theoretic viewpoint, assuming agents are selfish and rational, the equilibrium play
is desirable. However, in our model, we stipulate a deviation from the equilibrium play, towards a play that leads to a greater, but not necessarily maximum
payoff, in order to embed the notion of vulnerability of an agent. Thus, there
is no stipulation for an agent to adhere to a specific action set; if there were
one, we would not be able to embed the notion of vulnerability and uncertainty;
indeed, a stipulation of any kind would imply determinism, which runs contrary
to free will or choice of an agent. However, there are game theoretic constructs
like satisficing game theory [24,25], amongst others, that can model users (as opposed to automated agents). We mention such games in Section 6 on extensions
of this model; they are not considered in this paper. Herein, we shall use the
terms agent, user or player interchangeably; they imply the same entity unless
specifically mentioned otherwise.
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4
491
Trust Based Decision (TBD) Model
4.1
Base Scenario: Zero Trust
We now fix the base scenario, which is a 2-player game with a Nash Equilibrium.
Our notations closely follow standard game theoretic expositions as in [26]; although our models consider only 2 players, the notations are kept generic. The
trust game Gτ is defined by:
Gτ = {N, (Ai )i∈N , (ui )i∈N }
(1)
where N is the set of players, Ai is the action space of player i and ui is the payoff
function for player i, defined as ui : A → R, where A = ×i∈N Ai and R is the set of
real numbers. We assume that N and Ai are finite sets. Player i has at his disposal
the actions ai ∈ Ai . We denote the action spaces of all other players other than
i as A−i = ×j∈N \{i} Aj and a single element as a−i ∈ A−i . The repeated rounds
of the game Gτ are referred to as the supergame, which consists of a finite
sequence of the game Gτ , where the players choose the actions ai (t) ∈ Ai at
time instant t. For a sequence of k plays, we denote the history of player i by
Hi (k) = {ai (1), ai (2), . . . , ai (k)}∀ai (.) ∈ Ai and each element of Hi (k) by hi (k).
The payoff of player i at the end of any round is given by ui (ai , a−i ). The best
response action of player i is defined as bi (a−i ) = {ai ∈ Ai : ui (ai , a−i ) ≥
ui (a∗i , a−i )∀a∗i ∈ Ai }, i.e., given the plays of all the opponents a−i , the action
bi (a−i ) ensures the best payoff for player i. Hereafter, bi (a−i ) is denoted simply
as bi . We assume that the game’s payoff structure allows for (at least) a single
equilibrium point, at which the action profile of the players is (bi , b−i ). Thus
the single round payoff of the player i is given by ui (bi , b−i ). Intuitively, the
cumulative payoff of a sequence of k plays is k.ui (bi , b−i ). However, for the game
Gτ , similar to [26], we define the cumulative payoff to be a discounted one, where
the weights of the payoffs of older sequences are progressively lesser.
Definition 1. The discounted cumulative payoff in the game Gτ , of player i
over a sequence of k play’s, discounted by a factor of δ ∈ (0, 1), is defined as:
Ci (δ, k) = (1 − δ)
k
δ k−m ui (ai (m), a−i (m))
(2)
m=1
Unless otherwise specified, we shall denote Ci (δ, k) as Ci (δ). This formulation
places greater relevance to the most recent play (the k th play) and progressively
decreases the payoff of the past plays. From a trust game viewpoint, this is
intuitive; the closer δ is to 1, the greater the relevance to the most recent play
(due to the factor δ k−m ). Note that δ ∈ (0, 1) and does not ever assume the
value of 0 or 1. However, if we set δ = 0, Ci (δ) = 1; this can be interpreted as
setting no relevance at all to any of the plays, and hence the payoff incurred at
any stage is a constant: in the context of the model, setting no relevance to the
plays makes no sense and hence such a setting is invalid.1 In the base scenario,
1
We thank an anonymous reviewer for bringing out this point.
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the action taken by the players are the best responses to the action spaces of
other players; hence the cumulative payoff of player i is obtained when ai (.) = bi
and a−i (.) = b−i in Eq. 2. We denote this best response cumulative payoff as
Ci∗ (δ). The two values of per round payoff, ui (bi , b−i ) and cumulative discounted
payoff Ci∗ (δ) are used to refer to the Base Scenario with no trust.
4.2
Blind Trust
In this model, player i wants to initiate a trust relationship with the remaining
players. His goal is now to obtain the desired response from the remaining players
as opposed to the best response b−i played by the remaining players in the Base
Scenario.
Definition 2. For an action ai ∈ Ai , the desired response di (ai ) of player i is
defined as an action from the set A−i that increases the payoff of all players from
the equilibrium payoff, but does not provide the maximum possible payoff to all
the players other than possibly player i.
di (ai ) = {a−i ∈ A−i : ui (ai , a−i ) ≥ ui (bi , b−i ),
u−i (ai , b−i ) ≥ u−i (ai , a−i ) ≥ ui (bi , b−i )}
(3)
The desired response of player i is also denoted simply as di , where the action ai is understood to have been initiated. Note that di (ai ) ∈ A−i and is not
necessarily unique. Depending on the application domain, there may exist multiple di (ai ); however, their existence does not affect our model from a decision
theoretic viewpoint.
Definition 3. The index value τ of actions aτi ∈ Ai in a strictly increasing
ordering given by ui (a1i , di ) ≤ ui (a2i , di ) ≤ · · · ≤ ui (aτi , di ) ≤ · · · ≤ ui (aTi , di ) is
defined as the trust that player i places on the remaining players.
Note that the desired action di for aτi is not (necessarily) the same for all index
values. In situations where the context is clear, the symbol τ is used to represent
the trust of player i; in more generic terms, τ (i→-i) represents the cumulative
trust of player i on the remaining players, while τ (i→j ) represents the trust
of player i on player j. In the interest of keeping the formulation generic and
extensible, the notations of i and -i have been used; herein, we shall restrict
ourselves to N = 2, i.e., there are two players in the game Gτ ; thus i ∈ {1,2}
(-i denotes the ‘other’ player). In the formulations that follow, replacing i = 1
and -i = 2 represents the model for the two player scenario. The trust value τ
∈ {1, T}, where T is the maximum index. As a technical device, we may also
include a zero value in τ where the index value of zero is associated with the
best response action (and hence no trust).
In the basic blind trust scenario, player i assigns a trust value τ = 1, and
waits for the response from player -i . From the game theoretic viewpoint, this
game is a turn based game, where player -i knows the action taken by player i
before his turn to play. We call this model a blind trust model since there is no
prior communication between the two players for a contractual agreement on the
Towards Modeling Trust Based Decisions: A Game Theoretic Approach
493
action set, etc. Player i blindly trusts player -i and hopes for a reciprocation. At
this point, player -i may reciprocate by initiating the desired response di or the
best response b−i . The initiation of the desired response indicates the beginning
of a trust relationship.
From this basic blind trust Model, we wish to address several questions. First,
given that an agent (player i) wishes to initiate a trust relationship, what is
the vulnerability faced by player i? Secondly, assume that player i initiates a
blind trust relationship in the hopes of a future collaboration. Initially, player
-i may simply act in a ‘rational’2 manner and initiate the best response action,
but may eventually reconsider or ‘understand’ that player i wishes to initiate
a trust relationship. The reasons for the establishment and evolution of the
trust relationship are contextual and application/domain dependent, and are
hence outside the scope of this paper. However, the relevant question is: given
the amount of risk (maximum vulnerability) that player i is willing to tolerate,
what is the number of rounds of play that player i may expect to play? Towards
answering this question, we first define two types of risk and then evaluate the
expected number of rounds.
Instantaneous Per-Round and Cumulative Risk: The risk faced by a
player i are categorized into two types: the instantaneous per-round risk and
the cumulative k-stage risk.
Definition 4. The instantaneous per-round risk ρi of player i when initiating
action aτi with a trust τ on player -i is defined as the ratio of the difference
between the equilibrium payoff ui (bi , b−i ) and the best response ui (aτi , bi ) to the
equilibrium payoff.
ρi (τ ) = (1 −
ui (aτi , b−i )
)
ui (bi , b−i )
(4)
Note that, by the very definition of best response actions, ui (aτi , b−i ) ≤ ui (bi , b−i )
(otherwise, aτi = bi ). Thus, ρi (τ ) is the risk faced by the player i when trusting
player -i with a value of τ . Intuitively, the simplified cumulative k-stage risk is
k.ρi (τ ), assuming that the player -i plays the best response for all the k sequences
of the game.
Recall that we have defined a cumulative discounted payoff for the game Gτ
in definition 1, Eq. 2, i.e., the payoff of the k th round is discounted by a factor
(1 − δ). Towards this, a discounted cumulative k -stage risk is defined, similar
to definition 4. We first derive the discounted cumulative k -stage equilibrium
payoff and best response payoff. The discounted cumulative k-stage equilibrium
payoff of player i can be derived by substituting ai (m) = bi and a−i (m) = b−i
∀ m = {1,2,. . . ,k } in Eq. 2.
Cieq (δ) = (1 − δ)
k
δ k−m ui (bi , b−i ) = ui (bi , b−i )(1 − δ k )
(5)
m=1
2
‘Rational’ in the game theoretic sense, not in the context of the application domain.
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S. Vidyaraman, M. Chandrasekaran, and S. Upadhyaya
Note that the discounted equilibrium payoff is in fact the same as the equilibrium
payoff for any single round if δ →0, in which case, it is almost independent of
the number of rounds over which the game is played. Similarly, the discounted
cumulative k-stage best response payoff of player i can be derived by substituting
ai (m) = aτi and a−i (m) = b−i ∀ m = {1,2,. . . ,k } in Eq. 2.
Cibest (δ) = (1 − δ)
k
δ k−m ui (aτi , b−i ) = ui (aτi , b−i )(1 − δ k )
(6)
m=1
Definition 5. The discounted cumulative k-stage risk σi (τ, k) of player i over a
sequence of k plays of the game Gτ for the histories Hi (k) and H−i (k) is defined
as the ratio of the difference between the equilibrium payoff Cieq (δ) and the best
response payoff Cibest (δ) to the equilibrium payoff.
σi (τ, k) =
Cieq (δ) − Cibest (δ)
= ρi (τ )
Cieq (δ)
(7)
Usually, the cumulative risk is evaluated until player -i plays the desired action
di (aτi ).
Expected Number of Plays: Consider the situation when player i wishes to
initiate a trust relationship and initiates the action aτi instead of bi . Assume that
the maximum amount of risk the player i is willing to undertake is ri . We now
evaluate the number of rounds that player i may expect to play.
Consider the simplified cumulative k-stage risk, k.ρi (τ ). In this case, the maximum number of rounds player i can afford to play is kmax = ρir(τi ) . Now assume
that at any round of the k stages, the probability that player -i switches from
b−i to di is p. This probability is available to the player i through some context
specific mechanism, also called a belief in game theoretic literature. The probability that player -i switches from b−i to di at the k th round is p(1 − p)k−1 .
kmax
Thus the number of rounds player i may expect to play is m=1
mp(1 − p)m−1 .
Thus, considering the simplified cumulative k-stage risk, the number of rounds
player i may expect to play is:
E[k] =
1 − (kmax + 1)(1 − p)kmax + kmax (1 − p)kmax +1
p
(8)
where kmax = ρir(τi ) . Now, lets consider the discounted cumulative k-stage risk
σi (τ, k), which is equal to ρi (τ ). It can be observed trivially, that if the risks are
discounted for past rounds of the game, then player i may continue to play the
game infinitely if ri > ρi (τ ), just one round if ri = ρi (τ ) and may not play at all
if ri < ρi (τ ); i.e., the expected number of rounds is 1. Thus, for the discounted
case, the specification of the player’s absolute risk is not useful to determine the
maximum number of rounds. Instead, we define the discounted loss (in payoff)
for the player i for k rounds as:
Li (k) = Cieq (δ) − Cibest (δ) = (ui (bi , b−i ) − ui (aτi , b−i ))(1 − δ k )
(9)
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Let the loss that player i is willing to sustain in the trust initiative be li . Thus,
the maximum number of rounds is given by:
Li (kmax ) = li ⇒ kmax =
log(1 −
li
ui (bi ,b−i )−ui (aτi ,b−i ) )
(10)
log δ
Note that the loss li is not the absolute loss, but is the (maximum) an agent is
willing to sustain relative to the best response action b−i ; hence li < (ui (bi , b−i )−
ui (aτi , b−i ). Note also that kmax is inversely proportional to log δ, i.e., a players
maximum number of rounds depends on the extent to which he is willing to
discount past payoffs (or equivalently, losses). The expected number of rounds,
assuming that the probability that player -i switches from b−i to di is p, is given
by:
E[k] =
1 − (kmax + 1)(1 − p)kmax + kmax (1 − p)kmax +1
p
(11)
where kmax is given by Eq. 10.
Compensatory Update Strategy: Once the player -i responds with di , player
i may update the value of τ . Given that we are dealing with payoff values, we may
use an update strategy that assigns τ based on the payoff values. Our previous
work [27] describes a Compensatory Trust Model (CTM), where the trust value
may be updated as part of a compensation given to player -i based on his forgone
payoff. We briefly describe the intuition behind the update strategy based on the
CTM. When player -i initiates the desired action di at a particular round, denote
his loss of payoff as l−i = (u−i (aτi , b−i ) − u−i (aτi , di )), and the gain in payoff of
player i as gi = (ui (aτi , di ) − ui (aτi , b−i )). To ’share’ the loss and gain equally,
player i would have to transfer a payoff of l−i + 12 (gi − l−i ) to player -i. This
transfer may be made figuratively by updating the trust value proportionally,
for some δ ∈ (0,1), τ = δ(l−i + 12 (gi − l−i )), i.e., the trust update is proportional
to the discount δ of the payoffs. This is the ’compensation’ that player i pays
to player -i; hence the name Compensatory Update Strategy. This scheme can
be extended to include the risk faced by player i or the loss of payoff in a k stage game. As mentioned before, we do not present new trust assignment and
update methodologies; apart from the CTM update described above, any update
mechanism may be used to assign and update trust.
4.3
Incentive Trust
The incentive trust model considers the purpose of a trust transaction and models the scenario where agents may provide incentives to be trusted. Consider
the game Gτ with the players i and -i. The basic philosophy behind the trust
transaction, unlike the blind trust model, is the expectation of a return in trust
or value/payoff by a player. Towards this, we recast the game play described in
[28] to fit into our model. The incentive trust model is a game played over three
time periods. The play of the incentive trust model is described as follows:
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Initial Setup: Player -i wants player i to (a) trust him with a level of τ (i→i), hereafter denoted as τ and (b) initiate the action aτi . Towards this, player
-i states p−i , his stated probability that he will respond with the action di , the
desired action instead of b−i , the best response action. q−i is the true probability
that player -i will respond with the action di , the desired action instead of b−i ,
the best response action. In a similar vein, let pi be the stated probability (while
qi is the real probability) that player i will initiate action aτi . Both p−i and pi
are public knowledge.
Time Period 1:Player -i transfers a payoff of f (p−i ) as a support/proof of
his commitment to respond with the desired action di , where f (.) is a function
whose specification is to be determined.
Time Period 2: Player i, on receiving the support of f (p−i ) initiates the action
aτi .
Time Period 3: Player -i, in turn, initiates the desired action di .
The expected payoff of player i, w(p−i ) can be split into the following components:
1. Player i gets a payoff of f (p−i ) in time period 1.
2. Player i gets a payoff of only (1 − p−i )qi ui (aτi , b−i ) if player -i cheats and
responds with the best response b−i instead of the desired response.
3. Player i gets a payoff of qi p−i ui (aτi , di ) if he acts with action aτi and player
-i responds with the desired action.
Thus, the expected payoff of player i is given as:
w(p−i ) = f (p−i ) + qi {(1 − p−i )ui (aτi , b−i ) + p−i ui (aτi , di )}
(12)
Similarly, the expected payoff of player -i, z(pi ) can be split into the following
components:
1. If Player i initiates an action of aτi ,
(a) Player -i gets a payoff of pi (1 − q−i )u−i (aτi , b−i ) if he responds with the
best action b−i .
(b) Player -i gets a payoff of pi q−i u−i (aτi , di ) if he responds with the desired
action di .
2. If Player i does not initiate an action of aτi , then player -i loses a payoff of
(1 − pi )f (p−i ).
Thus, the expected payoff of player -i is given as:
z(pi ) = pi (1 − q−i )u−i (aτi , b−i ) + pi q−i u−i (aτi , di ) − (1 − pi )f (p−i )
(13)
Analysis: Consider the entire transaction and the purpose of the model: from
the viewpoint of player i, we would like to obtain an optimal guarantee f (p−i )
from player -i. Player -i, on the other hand, would like to ensure he at least
has a non-zero payoff from the entire transaction. Since Player i has the first
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497
position advantage, given p−i , he can fix the value of f (p−i ) that he receives from
player -i. The optimal value fopt (p−i ) is thus given by the first-order condition
w (p−i ) = 0.
fopt (p−i ) = p−i qi {ui (aτi , b−i ) − ui (aτi , di )} + c
(14)
where c is a constant such that c ≥ 0. If f (p−i ) satisfies Eq. 14, then player i
can expect a payoff of w(p−i ) = qi ui (aτi , b−i ) + c, independent of p−i . From the
viewpoint of player -i, the condition to be satisfied is z(pi ) ≥ 0. Thus, we derive
the condition for q−i , taking into account the value of f (p−i ) given in Eq. 14.
q−i ≥
pi (u−i (aτi , b−i )
pi u−i (aτi , b−i ) + (1 − pi )c
(15)
− u−i (aτi , di )) + (1 − pi )p−i (ui (aτi , b−i ) − ui (aτi , di ))
Note that q−i is not only dependent on pi , but also on p−i , on account of its
dependence on f (p−i ). Since player -i is the initiator in this model, it is his
responsibility to ensure a high enough value of q−i in order to ‘break even’ in
the transaction.
5
From Theory to Practice: Some Issues
The translation of game theoretic models to practical application scenarios is not
without its problems [29]. Although the blind trust and incentive trust models
capture the purpose of a trust transaction at an intuitive level, there are some
issues that should be addressed before its translation to practical scenarios. We
illustrate some of these issues and suggest directions for practical usage.
1. We have assumed that each user’s payoff is transferable to the other user;
e.g., in the incentive trust model player -i initially transfers an amount of
f (p−i ). Games where players can transfer their payoffs to others can be
modeled by a class of games called Transferable Utility (TU) games, which
also permit coalitions among groups of players. In practical scenarios, transferable payoffs have to be defined either in terms of monetary values or
resources/services/QoS-guarantees, etc., depending on the application domain.
2. We have not defined the semantics of the game which would indicate the
relevance of the equilibrium (or no trust) plays. The semantics of the game
must be defined for the particular application domain where the model is
applied to.
3. The most difficult part is the cohesion of the semantics of the game with
the zero trust interaction. In practical scenarios, zero trust usually implies
a lack of interaction and thus, no game plays at all. Bridging the game
semantics so that game plays occur in all scenarios is a challenging task;
in particular, the process of mechanism design must accurately map to the
scenarios so that the evaluation of iterated plays is relevant to the situation
under consideration. Of particular interest here is to model games which also
specify distrust.
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4. The specification or formulation of the utility functions is the first step towards enabling decision support based on trust. Depending on the application domain, utility functions may be formulated in functional form or,
if feasible, can be specified in discrete form for each action in the action
set. The works [30,31] provide methodologies for the construction of utility
functions depending on the history of an agent’s utility realization.
5. One of the pitfalls of game theoretic analysis is the existence of recursive
reasoning about the other agent’s knowledge [32]. The manifestation of this
pitfall is best illustrated with the incentive trust model. For example, player
i decides f (p−i ) based on the value of p−i , the stated probability of player
-i that he will play the desired action di instead of the best response b−i .
Although, there is a constant c ≥ 0 involved in Eq. 12 which gives player
i a leeway in choosing f (p−i ), it may be argued that player -i can quote a
low value of p−i and hence, the value of c must be chosen by player i, depending, amongst other factors, on player i’s belief in the value of p−i . Such
an analysis path usually leads to recursive reasoning and as illustrated by
experimental evidence [32,33,34], is not advisable in modeling what essentially is a subjective concept, beyond maybe two or three levels. Obviously,
if p−i < 12 , player i would not even consider entering into the transaction;
neither would player -i enter the transaction if f (p−i ) is too high.
6
Conclusion and Future Work
The purpose of most trust models and frameworks is to provide some form of decision support for an agent or user. Previous works in trust models have focused
on defining the notion of trust and evaluating it based on parametric representations, recommendations, etc. They provide decision support at an implicit level,
either through threshold based schemes or constraint satisfiability. This paper
provides a game theoretic approach to model trust based decisions. The model
takes a holistic view of a trust interaction by considering the ultimate purpose of
the transaction which may be spread over multiple periods. The very definition
of trust as an index into the agent’s action set provides decision support. Research in this direction has a potential to provide quantitative decision support
for trust frameworks with a multitude of actions choices. Future work on this
model comprise of two distinct paths: extensions and practical applications. Trust
decisions that are taken based on the recommendations of other players can be
incorporated into the model by means of an information structure. Furthermore,
the generic game Gτ assumes the existence of at least one equilibrium point; the
model can be improved by defining games with predefined payoff structures [35]
that imply the existence of a pure equilibrium point. Lastly, trust negotiations
that are made by people instead of automated agents have properties different
from automated agents. Satisficing game theory provides a mathematical basis
for user preferences/negotiations [24], which also account for the interests of
users in the welfare of other users.
Towards Modeling Trust Based Decisions: A Game Theoretic Approach
499
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